arX

iv:1

105.

4116

v1 [

astr

o-ph

.IM]

20 M

ay 2

011

A Fast Algorithm for Muon Track Reconstruction and its Application to theANTARES Neutrino Telescope

J.A. Aguilara, I. Al Samaraib, A. Albertc, M. Andred, M. Anghinolfie, G. Antonf, S. Anvarg, M. Ardidh, A.C. AssisJesusi , T. Astraatmadja1i, J-J. Aubertb, R. Auerf, B. Baretj , S. Basak, M. Bazzottil,m, V. Bertinb, S. Biagil,m, C.

Bigongiaria, C. Bogazzii, M. Bou-Caboh, M.C. Bouwhuisi, A.M. Brownb, J. Brunner2b, J. Bustob, F. Camarenah, A.Caponen,o, C.Carloganup, G. Carminatil,m, J. Carrb, S. Cecchinil,q, Ph. Charvisr, T. Chiarusil , M. Circellas, R.

Coniglionev, H. Costantinie, N. Cottinit, P. Coyleb, C. Curtilb, M.P. Decowskii, I. Dekeyseru, A. Deschampsr, C.Distefanov, C. Donzaudj,w, D. Dornicb,a, Q. Dorostix, D. Drouhinc, T. Eberlf, U. Emanuelea, J-P. Ernenweinb, S.

Escoffierb, F. Fehrf, V. Flaminioy,z, U. Fritschf, J-L. Fudau, S. Galatab, P. Gayp, G. Giacomellil,m, J.P.Gomez-Gonzaleza, K. Graff, G. Guillardaa, G. Halladjianb, G. Hallewellb, H. van Harenab, A.J. Heijboeri, Y. Hellor,

J.J. Hernandez-Reya, B. Heroldf, J. Hoßlf, C.C. Hsui , M. de Jong1i, M. Kadlerac, N. Kalantar-Nayestanakix, O.Kalekinf, A. Kappesf, U. Katzf , P. Kooijmani,ad,ae, C. Kopperf, A. Kouchnerj, V. Kulikovskiyaf,e, R. Lahmannf, P.

Lamareg, G. Larosah, D. Lefevreu, G. Limi,ae, D. Lo Prestiag, H. Loehnerx, S. Loucatost, F. Lucarellin,o, S.Manganoa, M. Marcelink, A. Margiottal,m, J.A. Martinez-Morah, A. Mazurek, A. Melif, T. Montarulis,ah, M.

Morgantiy,z, L. Moscosot,j, H. Motzf, C. Naumannt, M. Nefff, D. Palioselitisi , G.E.Pavalasai, P. Payreb, J. Petrovici,N. Picot-Clementeb, C. Picqt, V. Popaai, T. Pradieraa, E. Presanii, C. Raccac, C. Reedi , G. Riccobenev, C. Richardtf,

R. Richterf, A. Rostovtsevaj, M. Rujoiuai, G.V. Russoag, F. Salesaa, P. Sapienzav, F. Schockf, J-P. Schullert, R.Shanidzef, F. Simeonen,o, A. Spiessf, M. Spuriol,m, J.J.M. Steijgeri, Th. Stolarczykt, M. Taiutie,ak, C. Tamburiniu, L.Tascak, S. Toscanoa, B. Vallaget, V. Van Elewyckj, G. Vannonit, M. Vecchin,b, P. Vernint, G. Wijnkeri , E. de Wolfi,ae,

H. Yepesa, D. Zaborovaj, J.D. Zornozaa, J. Zunigaa

aIFIC - Instituto de Fısica Corpuscular, Edificios Investigacion de Paterna, CSIC - Universitat de Valencia, Apdo. deCorreos 22085, 46071 Valencia, SpainbCPPM - Centre de Physique des Particules de Marseille, Aix-Marseille Universite, CNRS/IN2P3, 163 Avenue de Luminy, Case 902, 13288 Marseille Cedex9, France

cGRPHE - Institut universitaire de technologie de Colmar, 34rue du Grillenbreit BP 50568 - 68008 Colmar, FrancedTechnical University of Catalonia,Laboratory of Applied Bioacoustics,Rambla Exposicio,08800 Vilanova i la Geltr´u,Barcelona, Spain

eINFN - Sezione di Genova, Via Dodecaneso 33, 16146 Genova, ItalyfFriedrich-Alexander-Universitat Erlangen-Nurnberg,Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany

gDirection des Sciences de la Matiere - Institut de recherche sur les lois fondamentales de l’Univers - Service d’Electronique des Detecteurs et d’Informatique, CEASaclay, 91191 Gif-sur-Yvette Cedex, France

hInstitut d’Investigacio per a la Gestio Integrada de Zones Costaneres (IGIC) - Universitat Politecnica de Valencia. C/ Paranimf 1. , 46730 Gandia, Spain.iNikhef, Science Park, Amsterdam, The Netherlands

jAPC - Laboratoire AstroParticule et Cosmologie, UMR 7164 (CNRS, Universite Paris 7 Diderot, CEA, Observatoire de Paris) 10, rue Alice Domon et Leonie Duquet75205 Paris Cedex 13, France

kLAM - Laboratoire d’Astrophysique de Marseille, Pole de l’Etoile Site de Chateau-Gombert, rue Frederic Joliot-Curie 38, 13388 Marseille Cedex 13, Francel INFN - Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy

mDipartimento di Fisica dell’Universita, Viale Berti Pichat 6/2, 40127 Bologna, ItalynDipartimento di Fisica dell’Universita La Sapienza, P.leAldo Moro 2, 00185 Roma, Italy

oINFN -Sezione di Roma, P.le Aldo Moro 2, 00185 Roma, ItalypLaboratoire de Physique Corpusculaire, IN2P3-CNRS, Universite Blaise Pascal, Clermont-Ferrand, France

qINAF-IASF, via P. Gobetti 101, 40129 Bologna, Italyr Geoazur - Universite de Nice Sophia-Antipolis, CNRS/INSU, IRD, Observatoire de la Cote d’Azur and Universite Pierre et Marie Curie, BP 48, 06235

Villefranche-sur-mer, FrancesINFN - Sezione di Bari, Via E. Orabona 4, 70126 Bari, Italy

tDirection des Sciences de la Matiere - Institut de recherche sur les lois fondamentales de l’Univers - Service de Physique des Particules, CEA Saclay, 91191Gif-sur-Yvette Cedex, France

uCOM - Centre d’Oceanologie de Marseille, CNRS/INSU et Universite de la Mediterranee, 163 Avenue de Luminy, Case 901, 13288 Marseille Cedex 9, FrancevINFN - Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, 95123 Catania, Italy

wUniv Paris-Sud , 91405 Orsay Cedex, FrancexKernfysisch Versneller Instituut (KVI), University of Groningen, Zernikelaan 25, 9747 AA Groningen, The Netherlands

yDipartimento di Fisica dell’Universita, Largo B. Pontecorvo 3, 56127 Pisa, ItalyzINFN - Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy

aaIPHC-Institut Pluridisciplinaire Hubert Curien - Universite de Strasbourg et CNRS/IN2P3 23 rue du Loess, BP 28, 67037 Strasbourg Cedex 2, FranceabRoyal Netherlands Institute for Sea Research (NIOZ), Landsdiep 4,1797 SZ ’t Horntje (Texel), The Netherlands

acDr. Remeis Sternwarte Bamberg, Sternwartstrasse 7,Bamberg,GermanyadUniversiteit Utrecht, Faculteit Betawetenschappen, Princetonplein 5, 3584 CC Utrecht, The Netherlands

aeUniversiteit van Amsterdam, Instituut voor Hoge-Energie Fysika, Science Park 105, 1098 XG Amsterdam, The NetherlandsafMoscow State University,Skobeltsyn Institute of Nuclear Physics,Leninskie gory, 119991 Moscow, Russia

agDipartimento di Fisica ed Astronomia dell’Universita, Viale Andrea Doria 6, 95125 Catania, ItalyahUniversity of Wisconsin - Madison, 53715, WI, USA

1Also at University of Leiden, the Netherlands2On leave at DESY, Platanenallee 6, D-15738 Zeuthen, Germany

1

aiInstitute for Space Sciences, R-77125 Bucharest, Magurele, RomaniaajITEP - Institute for Theoretical and Experimental Physics,B. Cheremushkinskaya 25, 117218 Moscow, Russia

akDipartimento di Fisica dell’Universita, Via Dodecaneso 33, 16146 Genova, Italy

Abstract

An algorithm is presented, that provides a fast and robust reconstruction of neutrino induced upward-going muonsand a discrimination of these events from downward-going atmospheric muon background in data collected by theANTARES neutrino telescope. The algorithm consists of a hitmerging and hit selection procedure followed by fittingsteps for a track hypothesis and a point-like light source. It is particularly well-suited for real time applications such asonline monitoring and fast triggering of optical follow-upobservations for multi-messenger studies. The performanceof the algorithm is evaluated with Monte Carlo simulations and various distributions are compared with that obtainedin ANTARES data.

1. Introduction

The main goal of neutrino telescope experiments such as AMANDA [1], IceCube [2], NT-200 in lake Baikal [3]and ANTARES [4] is the observation of high energy neutrinos from non-terrestrial sources. These instruments detectCherenkov light from the passage of relativistic charged particles produced in neutrino interactions in the detector orits surrounding material. By measuring the photon arrival times at known positions, the particle trajectory can be re-constructed. The flight direction of these particles is nearly colinear with the incident neutrino direction for neutrinosabove 10 TeV. The best angular resolution can be reached forνµ, νµ charged current interactions where the measuredparticle is a muon that can travel several kilometres in water at TeV energies. Events induced by neutrinos fromastrophysical sources must be distinguished from background which originates in the Earth’s atmosphere. Whereasatmospheric neutrinos are considered a non-reducible background (at least without the use of their energy), atmo-spheric muons can in principle be excluded by simple geometrical considerations. To exclude the contribution fromdownward-going atmospheric muons, it is sufficient to identify upward-going events, which can only be producedby neutrinos. This distinction requires a reliable determination of the elevation angle, because downward-going at-mospheric muons outnumber upward-going atmospheric neutrinos by 5 to 6 orders of magnitude for typical neutrinotelescope installation depths.

In the following, a tracking algorithm is presented which meets this goal. Apart from providing an excellent up-down separation, it is very fast and therefore well-qualified for real time applications. This has been demonstrated inANTARES, where the algorithm can cope with a trigger rate of 100 Hz, running on a single PC. It is used in variousonline monitoring tasks, for an analysis of atmospheric muons [5] and as an alert sending program to trigger opticalfollow-ups of selected neutrino events [6]. Other potential applications include the study of magnetic monopolesand nuclearites and the measurement of atmospheric neutrinos. The presented algorithm is complementary to otherreconstruction methods which have been developped in ANTARES [7, 8].

Section 2 describes the ANTARES detector, Section 3 explains the geometrical approximations which are used inall subsequent steps. The hit merging and hit selection methods are detailed in Sections 4 and 5 whereas the fittingprocedure can be found in Section 6. The performance of the entire algorithm introduced in Sections 2-6 is discussedin Section 7.

2. The ANTARES detector

The ANTARES detector is located about 40 km off the coast of Toulon, France, at a depth of 2475 m in theMediterranean Sea (42◦48′N, 6◦10′E). It consists of 12 flexible lines, each with a total height of 450 m, separated bydistances of 60-70 m from each other. They are anchored to thesea bed and kept near vertical by buoys at the topof the lines. Each line carries a total of 75 10-inch Hamamatsu photo multipliers (PMTs) housed in glass spheres,the optical modules (OM) [9], arranged in 25 storeys (three optical modules per storey) separated by 14.5 m, starting

Preprint submitted to Elsevier October 30, 2018

100 m above the sea floor. Each PMT is oriented 45◦ downward with respect to the vertical. A titanium cylinder ineach storey houses the electronics for readout and control.

The positions of the line anchors on the sea bed and the construction details (cable lengthetc.) of each detector lineare well known and stable in time. However the absolute positions and orientation of the individual OMs vary withtime as the lines move freely in the sea current. Their movements are monitored by a system of acoustic transpondersand receivers distributed over the lines and on the sea bed. After each cycle of acoustic signal exchange betweenthese elements, a triangulation provides the shape of each line. This is complemented by data from a compass cardwithin each electronics cylinder which measures the heading and the inclination of its storey. The processing of thesecombined data provides the position and orientation of eachOM with a precision of better than 10 cm, but after animportant delay.

3. Geometrical approximations

To qualify for real time applications the presented method uses only time independent geometrical informationsand the following approximations are applied:

Line shape Detector lines are considered to be perfectly vertical. Line distortions due to sea currents are ignored.

Storey geometry The detailed geometry of the storey is ignored. Each storey is considered as a single OM, which islocated directly on the detector line with an axis-symmetric field of view. The signals of the three PMTs withinone storey are combined.

These approximations are used in the hit merging, hit selection and fitting steps. Their effect on the angular resolutionis discussed in Section 7.3. Under the above approximations, a detector line is a straight line in space. It is alsoassumed that a muon track is a straight line,i.e. multiple scatteringetc. is ignored. Except for the special case thatthe muon track and a detector line are parallel (in the case ofan exactly vertical track), the point of closest approachof the muon track to the detector line can be determined. Mostof the Cherenkov light is expected to be seen in thevicinity of this point. This is used in the hit selection as well as in the fitting procedure of the algorithm describedbelow.

4. Hits and hit merging

The PMT signals are digitized by a custom built ASIC chip [10]. If the analog PMT signal crosses a presetthreshold, typically 1/3 photoelectron (p.e.), its arrival time is measured together with its charge. The latter is obtainedby integrating the analog PMT signal within a time window of 40 ns. Each such pair of time and charge is called ahit and the corresponding data for each hit are sent to shore.The data stream is processed by a computer farm in theshore station which searches for different physics signals according to predefined trigger conditions. If at least onetrigger pattern is found, all hits within a time window of fewµs around the trigger time are stored on disk. The DAQsystem is described in detail in [11].

For the data which will be presented in Section 7.3 two trigger algorithms are used. Both rely onL1 hitswhich aredefined either as 2 hits in coincidence within 20 ns in two OMs on the same storey or as a single hit with an amplitudelarger than a threshold of 3 or 10 p.e.. One of the used triggers requires at least five causally connected L1 hits withrespect to a given track direction. The second used trigger requires two pairs of L1 hits in adjacent or next-to-adjacentstoreys within 80 ns and 160 ns respectively.

Before starting the actual hit selection, time and charge calibrations are applied to transform raw data to physicalunits such as time (in nanoseconds) and charge (in equivalent number of photoelectrons). Up to now the standarddata handling in ANTARES has been described. The following steps are instead related to the presented algorithms.Different reconstruction methods might apply other hit treatment and hit selection procedures.

The calibrated hits are time-ordered for each storey. All hits on the same storey within a predefined time windoware merged, as applied to early ANTARES data in [7]. Merging is performed by adding the hit charges and keepingthe time of the earliest hit. The merging time window has to bechosen short enough to ensure a clean hit selection,with the earliest of the merged hits having a high probability of originating from a particle track. On the other hand,

3

the merging time window must be long enough to select most of the signal hits. For a 10-inch phototube, operatedat a threshold of 0.3 photoelectrons a typical noise rate of 60-100 kHz is observed in sea water [12]. In this case amerging window of 20 ns is appropriate. As a result, a list of merged hits per storey is obtained and is the basis for allsubsequent steps.

5. Hit selection

The light pattern expected from a highly relativistic charged particle is illustrated in Figure 1. The goal of the hitselection is to choose hits due to Cherenkov photons and avoid random hits from the optical background or scatteredlate hits. Late hits due to scattering are rare compared to typical optical background of 60-100 kHz. The probabilityof a Cherenkov photon atλ = 470 nm to suffer from light scattering on a path of 60 m (the inter-line distance andcoincidentally the water absorption length) is only few percent [13].

For tracks which are not parallel to a detector line, most of the Cherenkov light is expected near the point of closestapproach of the track to the detector line. Therefore, the first step of the hit selection consists of finding a “hot spot”of light on a detector line which originates from Cherenkov light induced by a passing particle. In the presence ofbackground light, a single high charge hit does not provide a“hot spot” with sufficient purity; two high charge hitsin adjacent or next-to-adjacent storeys are needed. The detector storeys are numbered consecutively along the z-axis,like floors in a building. For a given floori the adjacent or next-to-adjacent storeys are at floori ± j with j = 1, 2. Toensure that hits on floorsi andi ± j were caused by the same particle, the following condition onthe absolute hit timedifference∆t is imposed

∆t < j∆znc+ ts, (1)

with ∆z the absolute vertical distance between adjacent storeys,c the speed of light in vacuum,n the refractive indexof the medium andts an additional time delay which accounts for the timing uncertainty of the photon detectionitself and the simplifications of the detector geometry. It is assumed that storeys are equally spaced. Such a causalitycondition has been used for hit selection prior to track reconstruction in Baikal [3] and AMANDA [14]. For theANTARES detector with∆z = 14.5 m andts = 10 ns, one finds∆t < 80 ns for adjacent and∆t < 150 ns for next-to-adjacent storeys. Only merged hits which originate fromdifferent PMTs or with a charge above 2.5 photoelectronsare considered here, because they have a high probability ofbeing caused by a particle track. All hits which contributeto the above defined “hot spots” are stored as selected hits for the next steps. If several “hot spots” occur in the samestorey at different times, only the earliest of them is kept.

For the following steps, only detector lines having at leastone “hot spot” are considered. This condition removesall isolated hits from the sample. Using the “hot spot” hits as seeds, hits are added to the list of selected hits. Fromthis point on, the hit selection is exclusively based on timing information. No charge cut is applied at this stage. Thisallows to use the attenuated signal at larger distances fromthe point of closest approach as illustrated in Figure 1. Newhits on a given detector line are recursively sought within anarrow time window around the times of already selectedhits. The expected hit times in adjacent or next-to-adjacent storeys are calculated, assuming the hits arrive linearlyin thez-t plane as indicated by the asymptotes of the hyperbola in Figure 1. The projected arrival timeti± j at floori ± j, ( j = 1, 2) is related to any pair out of the three hit timesti , ti∓1, ti∓2 at floorsi, i ∓ 1, i ∓ 2 in the following way

ti± j = ti + j (ti − ti∓1) , (2)

ti± j = ti + j (ti − ti∓2) /2, (3)

ti± j = ti∓1 + ( j + 1) (ti∓1 − ti∓2) . (4)

In the following, time windows [tearlyi± j , t

latei± j ] based on the above equations are defined, which are used for the hit

selection. If only two out of the three hits in floorsi, i ∓ 1, i ∓ 2 exist, only one of the extrapolations above can beperformed. A new hit in floori ± j is accepted if it occurred within a time window of

tearlyi± j = ti± j − jts (5)

tlatei± j = max(ti + j∆z

nc+ ts, ti± j + jts), (6)

4

Time (ns)−1000 −500 0 500 1000 1500

Hei

ght (

m)

−200

−100

0

100

200

o90 o60 o30 o0

o90 o60

o30

o0

Time (ns)−1000 −500 0 500 1000 1500

Hei

ght (

m)

−200

−100

0

100

200o90 o60 o30 o0

o90o60

o30

o0

Figure 1: The pattern of Cherenkov light from a muon track; (left: upward-going;right: downward-going) on a single detector line. Verticalposition (z-coordinate) along the detector line and photonarrival time are related by a hyperbola (see Equations 17,18). The pointz = 0, t = 0defines the point of closest approach between track and detector line; here, the track passes at 10 m. The colours correspond to different trackangles: black 90◦ (vertical), red 60◦, green 30◦, blue 0◦ (horizontal). The thickness of the lines indicates the brightness of the expected signal.

with ts and∆z from Equation 1. This time window is asymmetric according tothe generic shape of the Cherenkovcone hyperbola in thez-t plane (see Figure 1 and Equations 17,18 discussed later). Ifall three hits exist, the minimumand maximumti± j are determined and a time window of

tearlyi± j = tmin

i± j − jts (7)

tlatei± j = tmax

i± j + jts, (8)

is used for floori ± j. If several hits in floori ± j meet the criteria, only the earliest one is selected. This procedure isapplied recursively. If new hits cannot be found in either floor i±1 or i±2, the procedure stops, avoiding gaps of morethan one floor. The occurrence of several well-separated “hot spots” on the same detector line serves as independentseeds to restart the procedure. The described hit selectionis performed once. It yields at most one hit per storey. Theresult of the hit selection on a bright neutrino candidate event from 2008 can be seen in Figure 3.

6. Fitting procedure

Before starting any fit, the list of selected hits is examined. Only events with more than 5 hits are accepted. If allselected hits are on a single detector line, a procedure for asingle-line fit is started, otherwise a multi-line fit procedureis performed. Both procedures are discussed in the following.

6.1. Fitting particle tracks

A particle track is considered to be a straight line in space.The particle is assumed to move with the speed of lightin vacuum. All space-time points,~p(t), that are part of the track can be parameterized as

~p(t) = ~q+ c(t − t0)~u. (9)

The particle passes through the point~q at timet0 (i.e. ~q = ~p(t0)) and moves in the direction~u. The vector~q can beshifted along the track by redefiningt0. Therefore the track is defined by a total of 5 parameters: three values to fix~qfor a given time and two angles to define~u:

~u = {cosθ cosφ, cosθ sinφ, sinθ} , (10)

5

Detector line

Track

Cherenkov cone

u

q,to

zcdc

p(tc)

Photondγ

z

Lx,Ly

θc

θγ

Figure 2: Illustration of the variables used to describe a track and its corresponding Cherenkov cone with respect to a vertical detector line.

with θ the elevation angle andφ the azimuth angle. The detector lines are approximated as vertical lines along thez-axis (see Section 2) at fixed horizontal positions (Lx, Ly). The z-component of the point of closest approach of aparticle track to a detector line is given by

zc =qz − uz(~q · ~u) + uz(Lxux + Lyuy)

1− u2z

, (11)

through which the particle passes at time

tc = t0 +1c

(Lxux + Lyuy + zcuz − ~q · ~u), (12)

at a distance

dc =

√

(px(tc) − Lx)2 + (py(tc) − Ly)2. (13)

The above-defined variables are illustrated in Figure 2. If the track is exactly vertical, and therefore parallel to thedetector line,tc = t0 andzc = qz are chosen. The track can be conveniently reparametrized interms ofzc, tc, dc for eachdetector line and the two angles which define~u. For a single-line fit the detector line can be placed at the coordinateorigin (Lx, Ly) = (0, 0) and Equations 11,12,13 simplify to

zc =qz− uz(~q · ~u)

1− u2z

, (14)

tc = t0 +qzuz − (~q · ~u)

c(1− u2z)

, (15)

dc =

√

p2x(tc) + p2

y(tc). (16)

6

A single detector line is invariant against a rotation of thecoordinate system around the z-axis. This means the track isnot fully defined and only 4 parameterszc, tc, dc anduz can be determined. No dependence from azimuth is observed.

To perform the track fit, it is necessary to know (a) the arrival time tγ of a Cherenkov photon at the detector linepositions (Lx, Ly, z), (b) its corresponding travel pathdγ and (c) its inclination with respect to the detector line, cosθγ.All three values can be derived from the parameters defined above and the refractive indexn which is related to theCherenkov angleθc by 1/n = cosθc:

tγ(z) = (tc − t0) +1c

(

(z− zc)uz +n2 − 1

ndγ(z)

)

, (17)

dγ(z) =n

√n2 − 1

√

d2c + (z− zc)2(1− u2

z), (18)

cosθγ(z) = (1− u2z)

z− zc

dγ(z)+

uz

n. (19)

These equations hold exactly for Cherenkov photons of a given wavelength. Dispersion and group velocity effects aswell as delays due to light scattering are ignored. An appropriate effective refractive index is used, depending on themedium in which the detector is installed.

6.2. Fitting point-like light sources: the bright-point fit

In contrast to a particle track, a ‘bright-point’ is defined as a point-like light source which emits a single light flashat a given moment. The light emission is assumed to be isotropic. The model of a bright-point does not only apply toartificial light sources such as optical beacons installed for detector calibrations [15] but also in some approximationto light from hadronic and electromagnetic showers, for which the actual extension of the shower is typically smallerthan the detector line distances. A bright-point is defined by four parameters: its position~q and its timet0. In analogywith the definitions of the point of closest approach for particle tracks, it is straightforward to see that for a bright-pointzc = qz, tc = t0 and

dc =

√

(qx − Lx)2 + (qy − Ly)2, (20)

which simplifies to

dc =

√

q2x + q2

y, (21)

for a single detector line at (Lx, Ly) = (0, 0). In this case, only the three parameterszc, tc, dc can be determined, whichagain means that the number of parameters is reduced by one. The photon arrival timetγ, its travel lengthdγ and itsangle with respect to a given arrival pointz along the detector line can thus be determined in analogy to the case of aparticle track:

dγ(z) =√

d2c + (z− qz)2, (22)

tγ(z) = t0 +nc

dγ, (23)

cosθγ(z) =(z− qz)

dγ. (24)

As in the treatment of particle tracks, it is assumed that allphotons have the same wavelength, which corresponds toa fixed effective refractive index.

6.3. Quality function

The quality function is based on the time differences between the hit timesti and the expected arrival timetγof photons from the track or bright-point, as in a standardχ2 fit. The quality function is extended with a term thataccounts for measured hit chargesai and the calculated photon travel distancesdγ. The full quality function is

Q =Nhit∑

i=1

(tγ − ti)2

σ2i

+A(ai)D(dγ)

〈a〉d0

, (25)

7

where〈a〉 is the average hit charge calculated from all hits which havebeen selected for the fit. The normalisationtermd0 and the functionsA(ai) andD(dγ) are discussed below. The second term of the right hand side of equation 25exploits the fact that an accumulation of storeys with high charges (hot spots) is expected on each detector line at itspoint of closest approach to the track. If such hot spots on several detector lines are arranged in a way that their z, tcoordinates indicate an upward-going pattern, the event has indeed a high probability to originate from an upward-going neutrino. This allows the isolation of a high purity neutrino sample. The concept is illustrated in Figure 3 wherethe arrangement of the “hot spots” on the various detector lines allows to classify this event as upward-going evenwithout an overlaid fitted track. The quality function depends on a number of parameters which have been tuned forsea water [12, 13] and the ANTARES detector [4].

The timing uncertainty,σi is set to 10 ns forai > 2.5 photoelectrons and to 20 ns otherwise. These values aresignificantly larger than the intrinsic transit time spreadof the ANTARES phototubes (1.3 ns). They take into accountan additional smearing due to the applied geometrical approximations from Section 3 and due to the occasionalpresence of late hits from small angle scattering or from electromagnetic processes which passed the hit selection.

The charge term of the quality function is not arranged as a difference between expected value and measuredcharge in order to avoid incorrect penalties from large expected charges. Instead, the chosen form gives a heuristicpenalty to the combination of high charge and large distance. The product is divided by the average charge of all theselected hits in an event,〈a〉, to compensate for the fact that more energetic tracks or showers will produce more lightat the same distance. The normalisationd0 balances the weight between the two terms. For ANTARES a value ofd0 = 50 m has been chosen, motivated by the fact that at this distance the typical signal in a photodetection unit whichpoints straight into the Cherenkov light front is of the order of one photoelectron.

The charges have to be corrected for the angular acceptance of the storey. Assuming a downward-looking hemi-spherical photodetection surface with a homogeneous efficiency for photodetection (a good approximation for anANTARES storey), the following simple correction functionfor the hit chargesa′i is obtained:

a′i =2ai

cosθγ + 1. (26)

The average charge〈a〉 is calculated from these corrected hit charges:

〈a〉 =1

Nhit

Nhit∑

1

a′i . (27)

Furthermore, charges are protected against extreme valuesby the following function:

A(ai) =a0a′i

√

a20 + a′2i

. (28)

The functionA(ai) introduces an artificial saturation such that fora′i ≪ a0, the charge is relatively unaffected,i.e.A(ai) ≈ a′i , whereas fora′i ≫ a0, the charge saturates atA(ai) ≈ a0, the chosen saturation value.

The photon travel distance is similarly protected using thefunction

D(dγ) =√

d21 + d2

γ, (29)

which introduces a minimal distanced1. For large distances,dγ ≫ d1, D(dγ) ≈ dγ, whereas for very small distances,dγ ≪ d1, D(dγ) ≈ d1. This avoids an excessive pull of the fitted trajectory towards the detector line.

Equations 28 and 29 can be motivated by the following consideration. The intensity of Cherenkov light decreaseslinearly with distance (ignoring absorption, dispersion and similar effects). Thus one can writea′i dγ = a0d1. ForANTARES, the constanta0d1 should be around 50 m×photoelectron, which corresponds to the observation that about50 photoelectrons can be measured for a minimum ionizing particle at 1 m distance, or equivalently 1 photoelectron isseen at a distance of 50 m. Using this identity, it follows that A(ai)D(dγ) = a′i dγ, i.e. the two functions have no effecton the product of charge times distance for Cherenkov light;however, for background light they produce the desiredpenalty.

8

6.4. Minimization procedure

The MIGRAD function of the MINUIT package [16] is used to determine the minimum of the quality function Q.The value ofQ/NDF (with NDF the number of degrees of freedom of the fit) after minimisation is calledfit qualityQ and is used for further selections. For the single-line bright-point fit, the parameters determined are the point ofclosest approach (i.e. zc, tc, dc) with an additional parameteruz in the case of a track fit. Equations 17-19 and 22-24are used to obtaindγ, tγ and cosθγ for particle tracks and bright-point fits, respectively.

The multi-line fits use the same set of equations. But in this case one more parameter is needed to evaluate them.For the multi-line track fit~q, uz andφ are used, for the multi-line bright-point fit~q andtc. Table 1 summarizes thenumber of fit parameters for the different cases.

single-line multi-lineparticle track 4 5bright-point 3 4

Table 1: Number of parameters for the different fits. Since multi-line events do not have rotational symmetry, an additional parameter is required.

A critical issue is the choice of the starting values and possible boundary conditions. As explained above, the setof fit parameters used for single-line bright-point fits iszc, tc, dc. Considering that most of the hits are grouped aroundthe point of closest approach,zc ≈ 〈zi〉 at tc ≈ 〈ti〉, the mean values of the selected hit positions and times are usedas starting values. The initial choice ofdc is not critical for the fit stability. It is set todc = 10 m. This completelydetermines the starting values for the single-line bright-point fit. For the single-line track-fit, a starting value foruz

has to be chosen as well. To avoid finding a wronguz, due to the inherent ambiguities when fitting Cherenkov lightcones, three different starting values ofuz are tried:−0.9, 0 and 0.9. The fit with the smallest fit qualityQ is taken.The allowed parameter range foruz is limited to physical values (i.e. −1 ≤ uz ≤ 1). Fit results for which the fit stopsat a boundary are excluded.

The multi-line bright-point fit starts from the center of gravity of the selected hits in space and time. The multi-linetrack fit requires, however, a prefit. For this purpose, an improved version [8] of the “DUMAND prefit” [17] is used,which fits a straight line to the hit positions while allowingfor a variable effective particle speed.

7. Test of the algorithm with Monte Carlo simulations and ANTARES data

The entire algorithm as described in Sections 3-6 was testedwith ANTARES data and simulations of atmosphericmuons and neutrinos.

7.1. Data sample

For the following analysis data are used which have been taken from December 2007 until end of 2008. Duringthis time a minimum of 9 detector lines was active. From May 2008 on the ANTARES detector was complete withits 12 lines. Only runs are used, which have at least 80% of active channels when averaged over the run. A channelis called active if it has an instantaneous counting rate of at least 40 kHz, well below the typical baseline noise rate of60 kHz. Further, runs are excluded for which more than 40% of the active channels have been inburst mode, whichmeans they measure an instantaneous counting rate more than20% above the usual baseline rate. This selectionprovides a data sample which corresponds to an active livetime of 173 days.

7.2. Simulations

Downward-going atmospheric muons were simulated with Corsika [18]. The primary flux was composed ofseveral nuclei according to [19] and the QGSJET hadronic model [20] was used in the shower development. Anaverage Mediterranean atmosphere is simulated,i.e. one which corresponds to a situation typically found in springor autumn. Seasonal variations of the muon flux due to atmospheric temperature and density variations are found tobe smaller than 3% in this region [21], and they are ignored inthis analysis. Upward-going neutrinos were simulatedaccording to the parameterization of the atmosphericνµ flux from [22] in the energy range from 10 GeV to 10 PeV.

9

Only charged current interactions ofνµ and νµ were considered. Disappearance of muon neutrinos due to neutrinooscillations was included in a simplified two-flavor model assuming maximal mixing and∆m2 = 2.4 · 10−3eV2.

The Cherenkov light, produced in the vicinity of the detector, was propagated taking into account light absorptionand scattering in sea water [13]. The angular acceptance, quantum efficiency and other characteristics of the PMTswere taken from [9] and the overall geometry corresponded tothe layout of the ANTARES detector [4]. The opticalnoise has been variable during the considered data taking period. It is simulated from counting rates observed inreal data. At the same time the definition of active and inactive channels has been applied from real data runs of theselected period. The generated statistics corresponds to an equivalent observation time of 10 years for atmosphericneutrinos and, depending on primary cosmic ray energy, from2 weeks up to 1 year for atmospheric muons.

A dedicated simulation has been done to study the impact of coincident atmospheric muon events. One monthof equivalent livetime has been simulated using the MUPAGE code [23]. An increase of 0.1% is found at triggerlevel. The fraction of downward reconstructed tracks increases by a similar fraction whereas for misreconstructedupward-going tracks an increase of 0.2% is found without using a cut in the fit quality.

7.3. Results

The accuracy in the reconstruction of the elevation angle (the angle with respect to the horizontal plane), whichneeds to be good for up-down separation of the reconstructedtracks, is illustrated in Figure 4. The reconstructedelevation angle is plotted versus the true elevation angle for down-going atmospheric muons and upward-going at-mospheric neutrinos, that are reconstructed as multi-lineevents. No further event selection was applied. More than95% of the fits converge for all events with selected hits on atleast two lines. Most events are located within a narrowband around the diagonal and 80% of the events have their elevation angle reconstructed to better than 5◦. No otherstructures are visible on the plots. It should be noted that this excellent agreement is obtained, even though more thanhalf of the triggered atmospheric muon events are muon bundles, whereas here just a single track reconstruction isperformed.

Figure 5 confronts data and the two Monte Carlo samples for the full 173 days livetime. The left plot of Figure 5shows the fit quality for the multi-line track fit of all the tracks reconstructed as upward-going. A cut on the trackfit quality can effectively suppress background from misreconstructed atmospheric muons while maintaining a goodefficiency for the atmospheric neutrino sample. A cut in the track fit quality of Q < 1.4 ensures a 90% purity whilekeeping 48% of the total sample of upward reconstructed multi-line neutrino tracks. The cut selects 665 neutrinocandidates in data (about 4 events per day), to be compared to609 events from the atmopheric neutrino Monte Carlosample and 40 misreconstructed downward-going atmospheric muon events. No event from the coincident muonsample survives the cut which gives a 90% upper limit of 13 such events in the final sample. Data and simulationagree within the shown statistical errors of the data forQ < 1.4, whereas forQ > 1.4 there is a systematic 30% excessof data. This excess is well within the estimated systematicerror of the atmospheric muon simulation (see below).The resulting elevation angle distribution of the selectedevents withQ < 1.4 is shown on the right plot of Figure 5.An excellent agreement between data and the atmospheric neutrino Monte Carlo sample based on the Bartol flux [22]is observed for the upward-going events, whereas the downward-going part features again a 30% excess of data withrespect to the atmospheric muon simulation. The grey band gives the systematic error of the simulation. An extensivediscussion of systematic effects is given in [5]. Main contributions are from the uncertainty in the effective area of thePMTs (10%), the uncertainty in the water absorption length (10%) and the uncertainty in the angular acceptance ofthe PMTs. The latter is better known close to the PMT axis (15%error) than in the backward hemisphere (up to 35%error) which leads to a larger total systematic error of 45% for downward-going atmospheric muons than for upward-going atmospheric neutrinos (32%). Within the quoted errors, data are compatible with the chosen flux models bothfor atmospheric neutrinos and for atmospheric muons.

Figure 6 compares data and simulation of some related quantities as the number of storeys used in the fit and thetotal charge of all hits used in the fit. Both plots show the comparison for the selected sampleQ < 1.4, which isdominated by atmospheric neutrinos. The shape of both distributions in data is well reproduced in the simulations.

Figure 7 shows the effective area for all reconstructed multi-line events as function of the true neutrino energy. Thecomparison of the solid and the dotted line indicates that more than 40% of the events pass the quality cut ofQ < 1.4for energies below 10 TeV. This fraction reduces to 20% at 100TeV and decreases further for higher energies. Thelow efficiency at highest energy can be explained by the use of a quality cut, which has been derived from Figure 5 for

10

Zenith : 27.2Fit on 8 line(s)

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 1

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 2

Run 37246 Frame 44741Wed Nov 19 05:28:37 2008 UTCTrigger bits 80002020 Line 1−12 Physics Trigger − Noisy channels treated

1 2 3 4 5 6 photons

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 5

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 3

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 4

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 6

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 9

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 7

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 8

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 10

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 11

time (nsec)−1000 −500 0 500 1000 1500 2000

heig

ht (

m)

050

100150200250300350400450500 Line 12

Figure 3: Event display of a bright neutrino event from the 2008 data taking period. The arrangement of the 12 histograms corresponds approxi-mately to the octagonal layout of the 12 detector lines on thesea floor. For each line the vertical position above the sea floor (m) is given as functionof the hit time (ns) (see also Figure 1).t = 0 corresponds to the time of the first hit which participated in the trigger. A window of−1000/+2000 nsis shown with respect to this time.z = 0 indicates the sea floor. Active detector elements are betweenz = 100 andz = 450 m. Circles denote hitswhich participated in the trigger, crosses are other hits, boxes indicate the hit selection of Section 5. Colors refer tothe hit charge as shown in thelegend on the top right part of the figure. The line width and style of the fitted track illustrates the minimal distance between track and detector line;thick and solid lines stand for closer distances than thin ordotted lines.

11

MCθsin0.4 0.5 0.6 0.7 0.8 0.9 1

Rec

oθ

sin

0.4

0.5

0.6

0.7

0.8

0.9

1

5000

10000

15000

20000

25000

30000

35000

40000

45000

MCθsin−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

Rec

oθ

sin

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Figure 4: Correlation between reconstructed and simulatedelevation angle;Left: down-going atmospheric muon events;Right: upward-goingatmospheric neutrino events.

Track Fit Quality0 0.5 1 1.5 2 2.5 3

Num

ber

of e

vent

s

1

10

210

310

410

Recoθsin

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Num

ber

of e

vent

s

1

10

210

310

410

510

610

Figure 5: Left: track fit quality for all upward reconstructed multi-line tracks for 2008 data (points with error bars) compared to a Monte Carlosample of downward-going atmospheric muons (dashed histogram) and upward-going atmospheric neutrinos (solid histogram); right: Elevationangle distribution for events withQ < 1.4. The grey band indicates the error band of the combined prediction from neutrino and muon simulations.

Number of Storeys5 10 15 20 25 30 35 40

Num

ber

of e

vent

s

−110

1

10

210

Amplitude (p.e.)50 100 150 200 250 300 350 400

Num

ber

of e

vent

s

−110

1

10

Figure 6: Left: Number of storeys andRight: total charge used in the track fit for all upward reconstructed multi-line tracks for events withQ < 1.4. Data (points with error bars) are compared to a Monte Carlosample of downward-going atmospheric muons (dashed histogram) andupward-going atmospheric neutrinos (solid histogram).

12

)ν

(E10

log1 2 3 4 5 6 7

)2 (

mef

fA

−610

−510

−410

−310

−210

−110

1

10

Figure 7: Effective area for all multi-line events (solid), selected events with Q < 1.4 (dotted) and selected events withQ < 1.3 + (0.04 · NDF)2

(dashed) as function of the true neutrino energy for upward-going neutrinos.

atmospheric neutrinos. By modifying the selection criteria one can easily recover the high energy part. The conditionQ < 1.3 + (0.04 · NDF)2 is used for the dashed line. It has the same performance asQ < 1.4 for Eν < 10 TeVwhere atmospheric neutrinos dominate, but a much improved performance atEν > 10 TeV. NDF = Nstorey− 5 ishere used as a simple estimator for the brightness and therefore energy of an event. This modified cut selects 624events in data to be compared to 588 atmopheric neutrinos and40 atmospheric muons keeping the quoted 90% purityof the neutrino sample. A full likelihood fit [8] has a similarperformance at high energies when requiring a 90%purity for atmospheric neutrinos. For energies below 1 TeV the method presented here is found more efficient thanthe likelihood fit.

|Recoθ−MCθ|0 1 2 3 4 5 6 7 8 9 10

Num

ber

of e

vent

s (a

.u.)

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

)datau•MCuacos(

0 2 4 6 8 10

Num

ber

of e

vent

s (a

.u.)

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

Figure 8: The distributions, normalized to the same content, of the angular error in degrees for selected atmospheric neutrinos (full) and atmosphericmuons (hatched);left: elevation angle error for multi-line events;right: space angle error for non-coplanar hit selection.

Figure 8 shows the angular resolution of the data sample which is selected by a cut of 1.4 on the fit quality.Both plots of Figure 8 are integrated over the full angular range and over the atmospheric neutrino energy spectrum.The elevation angle is reconstructed with a precision of 0.4◦ for upward-going neutrino-induced muons and 0.5◦ fordown-going muons (median of the angular error compared to the true muon direction). The space angle can only beunambiguously defined if the selected hits do not lie in a plane. Mirror solutions in azimuth are otherwise unavoidable.Because of the simplifications of the detector geometry usedduring reconstruction, a non-coplanar hit selection canonly be achieved when hits from at least three detector linesare used. This condition is fulfilled for 25% of the

13

selected multi-line atmospheric neutrino events. For thissub-sample a median space angle error of 1.0◦ is found foratmospheric neutrinos and 1.4◦ for atmospheric muons as shown on the right plot of Figure 8.

7.4. Improving the angular resolution

The angular resolution which has been quoted in the previoussection can be improved in several ways, withoutchanging the event selection and without compromizing the speed of the algorithm.

Resolving azimuthal degeneracyFor a particle trajectory reconstructed using hits on only two straight detector lines, there always exists an alterna-

tive trajectory having an identicalQ-value, but a different direction. The two trajectories will have the same elevations,but will differ in their azimuthal orientation. The degenerate trajectory is easily determined given the positions of thetwo detector lines and the trajectory resulting from the fit.First, the point at which the particle trajectory crosses theplane connecting the two detector lines is determined. The trajectory is then rotated about this point by an angle 2ψ

to form the degenerate track candidate, whereψ is the angle between the fit result and the plane connecting the twodetector lines.

Armed with both possible track solutions, the two candidates can be discriminated with the following procedure.The time residuals of all hits are calculated from the differences of the actual hit times and the theoretical hit timesaccording to Equation 17. The sum of the charge of each hit having a time residual smaller than 30 ns is thencalculated. This can include hits from any line and not just the two used in the original fit, helping to break thedegeneracy. The hit selection is done separately for both track candidates and the candidate with the largest totalcharge is chosen. In simulations of both downward-going muons and upward-going neutrinos this procedure choosesthe correct track candidate 70% of the time and improves the two-line angular resolution by 42% for upward-goingatmospheric neutrinos and by 15% for downward-going atmospheric muons.

Additional fit stepTaking the result of a multi-line track fit or its mirror solution as it was described in the previous sections as a

prefit, a new hit selection and a subsequent fit is performed. Every hit which has a time residual smaller than 20 nswith respect to the prefit track is chosen. These hits are thenused to minimize the function

M =Nhit∑

i=1

2

√

1+(tγ − ti)2

2σ2− 2

, (30)

which is a robust estimator that combines the properties of both least-squares and absolute-value minimizers [24].This estimator is quadratic for small values of the time residual (i.e. when |tγ − ti |/σ ≪ 1), just as with a standardχ2 estimator, but is linear for large time residuals. It is therefore not affected as dramatically by background hits withlarge residuals surviving the hit selection. Note that onlythe time residual of hits are minimized at this stage; noinformation concerning the number of photoelectrons detected is used. For the current studiesσ = 1 ns has beentaken. The precise value ofσ has little impact on the angular resolution.

Use of the detailed detector geometryWhile the algorithm has so far assumed a simplified detector geometry, it is possible to extend it in order to

exploit the actual location and orientation of the PMTs. This detailed geometry information can be employed in thecalculation of the predicted photon arrival time (tγ in Equation 30). The use of the resulting time residual for both thehit selection and track fit minimization, as described in theprevious section, has been studied. Simulations for thisstudy have been performed in the absence of sea currents,i.e. again for straight detector lines. The systematic angularshift due to the coherent inclination of all detector lines in a horizontal sea current is estimated to be smaller than 0.2◦

for a typical sea current of 5 cm/s as measured at the ANTARES site. However, such a systematicshift is not includedin the numbers and figures quoted below.

14

ResultsThe effect of these three additional steps can be seen in Table 2, which compares the median angular error of the

multi-line algorithm described in Section 6.1 to the subsequent fit using Equation 30 with and without the use of adetailed detector geometry. The condition on the number of detector linesN, from which hits have been used in thefit, is specified in the first column of Table 2. Results are given for upward-going atmospheric neutrinos as well asfor neutrinos from anE−2

ν flux, which is typical for astrophysical sources. Only events which fulfill the conditionQ < 1.3+ (0.04 · NDF)2 are included.

For tracks reconstructed using hits on only two lines in the initial fit, the improvement can be attributed mainly tothe partial resolution of the azimuthal symmetry. For the case where more than two lines had been used initially, themain improvement comes from the additional fit step. An additional 0.1◦ in resolution may be gained by using thedetailed detector geometry within each storey. The full multi-line sample (N > 1) profits from a combination of botheffects.

α for atmosphericν α for Φν ∝ E−2

lines Eq. 25 Eq. 30 Eq. 30 (geom) Eq. 25 Eq. 30 Eq. 30 (geom)N=2 4.2◦ 1.7◦ 1.5◦ 4.4◦ 1.3◦ 1.1◦

N>1 2.9◦ 1.3◦ 1.1◦ 1.3◦ 0.60◦ 0.52◦

N>2 1.00◦ 0.64◦ 0.54◦ 0.94◦ 0.47◦ 0.41◦

Table 2: The median angular error ( ˜α) in the multi-line fit algorithm as of Section 6.1 and the subsequent fit with and without using geometryinformation, for atmospheric neutrinos and neutrinos fromanE−2 flux.

The improvement in the angular error for neutrino-induced muons can also be seen in Figure 9 (left), which showsthe energy dependence of the median angular deviation from the true muon direction for all reconstructed tracksusing hits on more than 2 lines having aQ-value better than 1.3 + (0.04 · NDF)2. The selection is applied only tothe original track fit, so that the observed improvement of the angular resolution is the result a refined hit selection,combined with a new fit. The three curves correspond to the multi-line track fit from Section 6.1 (solid) and the fitsusing Equation 30 with an approximated geometry (dotted) and with a detailed geometry of each storey (dashed). Themuon track angular error is almost energy independent at neutrino energies above 10 TeV. It converges to the valuesquoted in table 2 for anE−2 flux.

This can be compared to the performance of a full likelihood fit [8] which uses the full geometry, a detailedmodelling of the time residual, (e.g. an energy dependent tail of late hits and a contribution of noise hits) and whichapplies a scan of starting positions of the fit to avoid local minima. Such a procedure can reach an angular resolutionof 0.3◦ at high energies for the entire multi-line sample, while using about 10 times more computing time than themethod presented here.

Figure 9 (right) shows the point spread function for neutrino events after the fit from Equation 30. Contrary to thefigures shown earlier the angular error is given here with respect to the neutrino direction - a quantity which is relevantto point back to a potential neutrino source. The optical followup program [6] uses a sub-sample of the events, shownon the dashed line in Figure 9 (right). As the field of view of the used robotic telescope is of the order of 2◦, it can beseen that more than 70% of the candidate sources would lay inside the field of view.

The improvement is further demonstrated by the time residual distributions, as shown in Figure 10. The timeresiduals are shown for all hits in each event for atmospheric neutrino events. The residuals of the subsequent fit usingthe detailed geometry show a more pronounced peak at zero, whose width is roughly 40% smaller compared to themulti-line fit from Section 6.1.

These extensions to the algorithm do not significantly affect the time needed for track reconstruction. Includingthe subsequent fit, the entire reconstruction algorithm takes an average of 5 ms per muon event on an Intel Xeon2.66 GHz CPU for events recorded by the 12-line ANTARES detector.

8. Conclusions

A generic and fast track fit algorithm for muon tracks in neutrino telescopes has been presented which can reliablydistinguish upward-going neutrinos from the overwhelmingbackground of downward-going muons. The method is

15

(GeV)νE10

Log1 2 3 4 5 6

(de

g)α∼

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Angular error (degree)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Fra

ctio

n of

eve

nts

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 9: Left: Dependence of the angular resolution for selected tracks (Q < 1.3 + (0.04 · NDF)2,N > 2) on Eν . The median angular errorof the multi-line fit (solid line) is compared against the fit from Equation 30 without (dotted) and with (dashed) use of a detailed description ofthe geometry of a storey;right: Cumulative point spread function for selected upward-going neutrino events with anE−2 spectrum andQ <1.3+ (0.04 · NDF)2,N > 2 for multi-line fits (solid) and fits from Equation 30 with an approximate geometry (dotted).

used in ANTARES for different applications such as an online event display, a neutrino monitor and an alert sendingprogram to trigger optical follow-ups of selected neutrinoevents. It has also been used for an analysis of atmosphericmuons [5]. Other potential applications include the study of magnetic monopoles and nuclearites and the measurementof atmospheric neutrinos.

9. Acknowledgement

The authors acknowledge the financial support of the fundingagencies: Centre National de la Recherche Sci-entifique (CNRS), Commissariat a l’Energie Atomique et auxEnergies Alternatives (CEA), Agence National de laRecherche (ANR), Commission Europenne (FEDER fund and Marie Curie Program), Region Alsace (contrat CPER),Region Provence-Alpes-Cte d’Azur, Departement du Var and Ville de La Seyne-sur-Mer, France; Bundesministeriumfur Bildung und Forschung (BMBF), Germany; Istituto Nazionale di Fisica Nucleare (INFN), Italy; Stichting voorFundamenteel Onderzoek der Materie (FOM), Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO),the Netherlands; Council of the President of the Russian Federation for young scientists and leading scientific schoolssupporting grants, Russia; National Authority for Scientific Research (ANCS), Romania; Ministerio de Ciencia e In-novacion (MICINN), Prometeo of Generalitat Valenciana and MultiDark, Spain. Technical support of Ifremer, AIMand Foselev Marine for the sea operation and the CC-IN2P3 forthe computing facilities is acknowledged.

References

[1] J. Ahrenset al. [AMANDA Collaboration], Astrophys. J.583 (2003) 1040.[2] R. Abbasiet al. [IceCube Collaboration],Astrophys. J.701 (2009) 47.[3] V. A. Balkanovet al. [Baikal Collaboration],Astropart. Phys.12 (1999) 75.[4] M. Ageronet al. [ANTARES Collaboration],Astropart. Phys.31 (2009) 277.[5] J.A. Aguilar et al., [ANTARES Collaboration],Astropart. Phys.34 (2010) 179.[6] D. Dornic et al., arXiv:0810.1416 [astro-ph].[7] R. Bruijn, PhD thesis, University of Amsterdam, March 2008.

( http://www.nikhef.nl/pub/services/biblio/theses_pdf/thesis_R_Bruijn.pdf )[8] A. Heijboer, PhD thesis, University of Amsterdam, June 2004, p.57ff.

( http://www.nikhef.nl/pub/services/biblio/theses_pdf/thesis_A_Heijboer.pdf )[9] P. Amramet al., [ANTARES Collaboration],Nucl. Instrum. Meth. A484 (2002) 369.

[10] J.A. Aguilaret al., [ANTARES Collaboration],Nucl. Instrum. and Meth.A 622 (2010) 59.

16

(ns)pred - thitt-30 -20 -10 0 10 20 30

Num

ber

of h

its (

a.u.

)

-710

-610

Figure 10: Time residuals of all hits (not just those used in the fit) for the multi-line track fit (full) and the subsequent fit using geometry information(hatched).

[11] J.A. Aguilaret al., [ANTARES Collaboration],Nucl. Instrum. Meth.A 570 (2007) 107.[12] P. Amramet al., [ANTARES Collaboration],Astropart. Phys.13 (2000) 127.[13] J.A. Aguilaret al., [ANTARES Collaboration],Astropart. Phys.23 (2005) 131.[14] J. Ahrenset al. [AMANDA Collaboration], Nucl. Instrum. Meth.A 524 (2004) 169.[15] M. Ageronet al., [ANTARES Collaboration],Nucl. Instrum. Meth.A 578 (2007) 498.[16] F. James and M. Roos,Comput. Phys. Commun.10 (1975) 343.[17] V. Stenger, DUMAND External report HDC-1-90, 1990.[18] D. Hecket al., Report FZKA 6019 (1998), Forschungszentrum Karlsruhe;

D. Heck and J. Knapp, Report FZKA 6097 (1998), Forschungszentrum Karlsruhe.[19] S.I. Nikolskyet al., Sov. Phys.JETP60 (1984) 10;

E.V. Bugaevet al., Phys. Rev.D58 (1998) 05401.[20] N.N. Kalmykov and S.S. Ostapchenko,Yad. Fiz.56 (1993) 105;

N.N. Kalmykov and S.S. OstapchenkoPhys. At. Nucl.56(3) (1993) 346.[21] M. Ambrosioet al., [MACRO Collaboration],Astropart. Phys.7 (1997) 109;

Y. Becheriniet al., [MACRO Collaboration],Astropart. Phys.23 (2005) 341.[22] G. Barret al., Phys. Rev.D39 (1989) 3532; V. Agrawalet al., Phys. Rev.D53 (1996) 1314.[23] G. Carminatiet al., Comput. Phys. Commun.179 (2008) 915.[24] Z. Zhang,Image and Vision Computing Journal15 (1997) 59.

17